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Mirrors > Home > MPE Home > Th. List > Mathboxes > 6gbe | Structured version Visualization version GIF version |
Description: 6 is an even Goldbach number. (Contributed by AV, 20-Jul-2020.) |
Ref | Expression |
---|---|
6gbe | ⊢ 6 ∈ GoldbachEven |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 6even 46365 | . 2 ⊢ 6 ∈ Even | |
2 | 3prm 16627 | . . 3 ⊢ 3 ∈ ℙ | |
3 | 3odd 46362 | . . . 4 ⊢ 3 ∈ Odd | |
4 | gbpart6 46420 | . . . 4 ⊢ 6 = (3 + 3) | |
5 | 3, 3, 4 | 3pm3.2i 1339 | . . 3 ⊢ (3 ∈ Odd ∧ 3 ∈ Odd ∧ 6 = (3 + 3)) |
6 | eleq1 2821 | . . . . 5 ⊢ (𝑝 = 3 → (𝑝 ∈ Odd ↔ 3 ∈ Odd )) | |
7 | biidd 261 | . . . . 5 ⊢ (𝑝 = 3 → (𝑞 ∈ Odd ↔ 𝑞 ∈ Odd )) | |
8 | oveq1 7412 | . . . . . 6 ⊢ (𝑝 = 3 → (𝑝 + 𝑞) = (3 + 𝑞)) | |
9 | 8 | eqeq2d 2743 | . . . . 5 ⊢ (𝑝 = 3 → (6 = (𝑝 + 𝑞) ↔ 6 = (3 + 𝑞))) |
10 | 6, 7, 9 | 3anbi123d 1436 | . . . 4 ⊢ (𝑝 = 3 → ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 6 = (𝑝 + 𝑞)) ↔ (3 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 6 = (3 + 𝑞)))) |
11 | biidd 261 | . . . . 5 ⊢ (𝑞 = 3 → (3 ∈ Odd ↔ 3 ∈ Odd )) | |
12 | eleq1 2821 | . . . . 5 ⊢ (𝑞 = 3 → (𝑞 ∈ Odd ↔ 3 ∈ Odd )) | |
13 | oveq2 7413 | . . . . . 6 ⊢ (𝑞 = 3 → (3 + 𝑞) = (3 + 3)) | |
14 | 13 | eqeq2d 2743 | . . . . 5 ⊢ (𝑞 = 3 → (6 = (3 + 𝑞) ↔ 6 = (3 + 3))) |
15 | 11, 12, 14 | 3anbi123d 1436 | . . . 4 ⊢ (𝑞 = 3 → ((3 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 6 = (3 + 𝑞)) ↔ (3 ∈ Odd ∧ 3 ∈ Odd ∧ 6 = (3 + 3)))) |
16 | 10, 15 | rspc2ev 3623 | . . 3 ⊢ ((3 ∈ ℙ ∧ 3 ∈ ℙ ∧ (3 ∈ Odd ∧ 3 ∈ Odd ∧ 6 = (3 + 3))) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 6 = (𝑝 + 𝑞))) |
17 | 2, 2, 5, 16 | mp3an 1461 | . 2 ⊢ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 6 = (𝑝 + 𝑞)) |
18 | isgbe 46405 | . 2 ⊢ (6 ∈ GoldbachEven ↔ (6 ∈ Even ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 6 = (𝑝 + 𝑞)))) | |
19 | 1, 17, 18 | mpbir2an 709 | 1 ⊢ 6 ∈ GoldbachEven |
Colors of variables: wff setvar class |
Syntax hints: ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∃wrex 3070 (class class class)co 7405 + caddc 11109 3c3 12264 6c6 12267 ℙcprime 16604 Even ceven 46278 Odd codd 46279 GoldbachEven cgbe 46399 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-sup 9433 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-n0 12469 df-z 12555 df-uz 12819 df-rp 12971 df-fz 13481 df-seq 13963 df-exp 14024 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-dvds 16194 df-prm 16605 df-even 46280 df-odd 46281 df-gbe 46402 |
This theorem is referenced by: nnsum3primesle9 46448 |
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